Integrally convex set

An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry.

Denote by ch(X) the convex hull of X.

Its convex hull ch(X) contains, for example, the point y = (1.2, 0.5).

[1] In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex.

Iimura, Murota and Tamura[3] have shown the following property of integrally convex set.

be a finite integrally convex set.

There exists a triangulation of ch(X) that is integral, i.e.: The example set X is not integrally convex, and indeed ch(X) does not admit an integral triangulation: every triangulation of ch(X), either has to add vertices not in X, or has to include simplices that are not contained in a single cell.

In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex, and indeed admits an integral triangulation, e.g. with the three simplices {(0,0),(1,0),(1,1)} and {(1,0),(2,0),(2,1)} and {(1,0),(1,1),(2,1)}.